7. POLARIZATION AND PHYSICAL EFFECTS

What is microwave background polarization good for? One basic
and model-independent answer to this question was outlined
above: polarization can provide a clean demonstration of the
existence of acoustic oscillations in the early universe. The
fact that three of the six polarization-temperature power spectra
are zero for linear scalar perturbations gives several other
interesting and model-independent probes of physics.

The most important is that the ``curl'' polarization power
spectrum directly reflects the existence of any vector
(vorticity) or tensor (gravitational wave) metric perturbations.
Inflation models generically predict a nearly-scale invariant
spectrum of tensor perturbations, while defects or other
active sources produce significant amounts of both vector
and tensor perturbations. If the measured temperature power
spectrum of the microwave background turns out to look
different than what is expected in the broad class of inflation-like
cosmological models, polarization will tell what part
of the temperature anisotropies arise from vector and
tensor perturbations. More intriguingly, in inflation models,
the amplitude of the tensor perturbations is directly
proportional to the energy scale at which inflation occurred, so
characterizing the gravitational wave background becomes a
probe of GUT-scale physics at 1016 GeV! Inflation also
predicts potentially measurable relationships between the
amplitudes and power law indices of the primordial density
and gravitational wave perturbations (see
(Lidsey et
al., 1997)
a comprehensive overview), and measuring a
ClC power spectrum appears to be the only way to
obtain precise enough measurements of the tensor perturbations
to test these predictions. A microwave background map with
forseeable sensitivity could measure gravitational wave
perturbations with amplitudes smaller than 10-3 times
the amplitude of density perturbations
(Kamionkowski and Kosowsky, 1998),
thanks to the fact
that the density perturbations don't contribute to
ClC.
The tensor perturbations generally contribute significantly
to the temperature perturbations at angular scales larger
than two degrees (l 100) in a flat
universe
but have a much broader range of scales in polarization
(50 l 500).
For tensor and vector
perturbations, the amplitude of the C-polarization
is generally about the same as that of the G-polarization;
if the perturbations inducing the COBE temperature anisotropies are
10% tensors, then we expect the peak of
l ClC 10-15 at
angular scales around l = 80.
An experimental challenge not for the faint of heart!

A second source of C-type polarization is gravitational
lensing. The mass distribution in the universe between us and
the surface of last scatter will bend the geodesics of the
microwave background photons. This lensing can be described by
an effective displacement field, in which the temperature and
polarization at each point of the sky in an unlensed universe
is mapped to a nearby but different point on the sky when lensing
is accounted for. The displacement alters the shape of temperature
contours in the microwave background, and likewise distorts
the polarization pattern, inducing some curl component to the
polarization field. Detailed calculations of this
effect and the induced ClC have been made by
Zaldarriaga
and Seljak (1998). The amplitude of this effect
is expected to be around l2ClC 10-14 on a broad
range of subdegree
angular scales (200 l 3000) with the power spectrum
peaking around l = 1000 in a flat universe. This lensing polarization
signal is just at the limit of detectability for the upcoming
Planck satellite; future polarization satellites with better
sensitivity could make detailed lensing maps based on the curl
component of microwave background polarization.
It is interesting to note that tensor perturbations and gravitational
lensing are substantially distinguishable by their different
angular scales. Note that the most recent version of the
publicly available CMBFAST code by Seljak and Zaldarriaga
(Seljak and
Zaldarriaga, 1996) computes polarization from both
tensor modes and from gravitational lensing.

A third source of C-type polarization is a primordial magnetic
field. If a magnetic field was present at recombination,
the linear polarization of electromagnetic radiation would
undergo a Faraday rotation as it propagated through the
surface of last scatter while significant numbers of free
electrons were still present. (Such rotation could also
occur after reionization, but both the electron density
and the field strength would be much smaller and the
resulting rotation is small compared to the primordial signal).
This effect rotates an initial G-type polarization field
into a C-type polarization field. A detailed estimate of
the magnitude of this effect
(Kosowsky
and Loeb, 1997) shows
that a primordial field with present strength 10-9 gauss
induces a measurable one-degree rotation in the polarization
at a frequency of 30 GHz. Faraday rotation depends quadratically
on wavelength of the radiation, so down at 3 GHz, the rotation would
be a huge 100 degrees (although the polarized emission from
synchrotron radiation would also be correspondingly larger).
Such a rotation will induce
l2CC at a level of
between 10-15 for one degree of rotation and 10-11
for large rotations. Additionally, it has been pointed out that
Faraday rotation will contribute also to the
ClTC
cross-correlation at corresponding levels
(Scannapieco
and Ferreira, 1997)
as well as to ClGC.
Investigation of the angular dependence
and detectability of such a signal is ongoing
(Mack and
Kosowsky, 1999).
The best current constraints on a homogeneous component of a primordial
magnetic field come from COBE constraints on anisotropic Bianchi
spacetimes
(Barrow,
Ferreira and Silk, 1997), because a universe which contains
a homogeneous magnetic field cannot be statistically isotropic.
Detection of a significant
primordial magnetic field would both provide the
seed field needed to generate current galactic and subgalactic-scale
magnetic fields via the dynamo mechanism, and also provide a very
interesting constraint on fundamental particle physics, particularly
if a field on large scales is detected (see, e.g.,
Turner and
Widrow (1988) or
Gasperini
et al. (1995)).

Faraday rotation from magnetic fields is a special case
of cosmological birefringence: rotation of
polarization by differing amounts depending on direction
of observation. Such rotation could arise from
interactions between photons and other unknown fields.
Constraints on the C-polarization of the microwave background
could strongly constrain new pseudoscalar particles (see, e.g.,
Carroll and
Field (1997)).
More generally, non-zero cosmological contributions to
the ClTC and
ClGC cross correlations,
which must be zero if parity is a valid symmetry of
the cosmological perturbations, would indicate some
intrinsic parity to either the primordial perturbations
(Lue,
Wang, and Kamionkowski, 1998)
or to some interaction of the microwave background photons
(Carroll,
1998). These types of effects are generally
independent of photon frequency, so they can be distinguished
from Faraday rotation through microwave background frequency dependence.

The above signals are all model-independent probes of new physics
using microwave background polarization. An additional less
daring but initially more useful and important use of polarization
is in determining and constraining the basic background
cosmology of the universe. It has been appreciated for
several years now that the microwave background offers the
cleanest and most powerful constraint on the gross features of the universe
(Jungman et
al., 1996).
If the universe is
described by an inflation-type model, with nearly scale-invariant
initial adiabatic perturbations which evolve via gravitational
instability, then the power spectrum of microwave background
temperature fluctuations can strongly constrain nearly all
cosmological parameters describing the universe: densities
of various matter and energy components, amplitudes and
power laws of initial density and gravitational wave perturbations,
the Hubble parameter, and the redshift of reionization. More
recent work
(Eisenstein,
Hu and Tegmark, 1999;
Zaldarriaga,
Seljak, and Spergel, 1997) has shown
that
the addition of polarization information can help tighten
these constraints considerably, mainly because the new information
now gives four theoretical power spectra to match
instead of just one.
Polarization particularly helps constrain the reionization
redshift and the baryon density
(Zaldarriaga
and Harari, 1995).
Polarization will also be important for deciphering the
universe if measurements of the temperature anisotropies reveal
that the universe is not described by the simple
class of inflation-like cosmological models: it is a strong
discriminator between vector and tensor perturbations and
scalar perturbations
(Kamionkowski, Kosowsky, and Stebbins, 1997).

Finally, no discussion of this sort would be completely
honest without mentioning the thorny issue of foreground emission.
We are gradually concluding that foregrounds have some
non-negligible effect on temperature anisotropies, but that
the amplitudes of various foregrounds are small enough that
they will not substantially hinder our ability to draw
cosmological conclusions from microwave background temperature
maps (see, e.g.,
Tegmark (1998)
for a recent estimate).
Whether the same will prove true for polarization is
unknown at present. Free-free emission is likely to have only
negligible polarization, but synchrotron emission will be
strongly polarized, and the polarization of dust emission is
difficult to estimate reliably
(Draine and
Lazarian, 1998).
Polarized emission from
radio point sources is another potential problem.
No present measurements have
had sufficient sensitivity to detect polarized emission from
any of these foreground sources, so it is difficult to predict
the foreground impact. My own guess is that the G-polarization
component, from which acoustic oscillations can be confirmed and
from which parameter estimation can be significantly improved,
will face foreground contamination comparable to
the temperature anisotropies. If so, and if the polarization
foregrounds are divided evenly between C and G polarization
components, then control of foregrounds will become crucial
for the very interesting physics probed by the cosmological
C polarization. But I fully
expect that through a combination of techniques, including
carefully tailored sky cuts, measurements at many frequencies,
improved theoretical understanding,
foreground nongaussianity, and
foreground template matching,
we will separate out the small cosmological polarization
signals from whatever polarized foregrounds are out there.

The next five years will bring us microwave background temperature
maps of vastly improved sensitivity and resolution, and almost
certainly the first detection of microwave background polarization.
These observations will provide us with very tight constraints
on our cosmological model, or else will reveal some new and
unexpected aspect of our universe. Either way, the microwave
background will be the cornerstone of a mature cosmology. What is
left to do after Planck? One good answer to this question, I believe,
is very high
sensitivity measurements of microwave background polarization.
Such observations hold the promise of probing the potential
driving inflation, detecting primordial magnetic fields,
mapping the matter distribution in the universe, and likely
a variety of other interesting physics yet to be explored.

This work has been supported by the NASA Theory Program.
Portions of this work were
done at the Institute for Advanced Study.